Definitions

Classical

Classically, a monoid is a set MM equipped with a binary operation μ:M×M→M\mu: M \times M \to M and special element 1∈M1 \in M such that 11 and x⋅y=μ(x,y)x \cdot y = \mu(x,y) satisfy the usual axioms of an associative product with unit, namely the associative law:

As a one-object category

Equivalently, and more efficiently, we may say that a (classical) monoid is the hom-set of a category with a single object, equipped with the structure of its unit element and composition.

More tersely, one may say that a monoid is a category with a single object, or more precisely (to get the proper morphisms and 22-morphisms) a pointed category with a single object. But taking this too literally may create conflicts in notation. To avoid this, for a given monoid MM, we write BM\mathbf{B}M for the corresponding category with single object •\bullet and with MM as its hom-set: the delooping of MM, so that M=HomBM(•,•)M = Hom_{\mathbf{B}M}(\bullet, \bullet). This realizes every monoid as a monoid of endomorphisms.

Similarly, a monoid in (C,⊗,I)(C,\otimes,I) may be defined as the hom-object of a CC-enriched category with a single object, equipped with its composition and identity-assigning morphisms; and so on, as in the classical (i.e. Set\mathbf{Set}-enriched) case.

Category of monoids

Examples

A monoid in which every element has an inverse is a group. For that reason monoids are often known (especially outside category theory) as semi-groups. (But this term is often extended to monoids without identities, that is to sets equipped with any associative operation.)

A monoid object in Ab (with the usual tensor product of ℤ\mathbb{Z}-modules as the tensor product) is a ring. A monoid object in the category of vector spaces over a field kk (with the usual tensor product of vector spaces) is an algebra over kk.

For a commutative ring RR, a monoid object in the category of RR-modules (with its usual tensor product) is an RR-algebra.

For any category CC, the endofunctor category CCC^C has a monoidal structure induced by composition of endofunctors, and a monoid object in CCC^C is a monad on CC.

These are examples of monoids internal to monoidal categories. More generally, given any bicategoryBB and a chosen object aa, the hom-categoryB(a,a)B(a,a) has the structure of a monoidal category. So, the concept of monoid makes sense in any bicategoryBB: we define a monoid in BB to be a monoid in B(a,a)B(a,a) for some object a∈Ba \in B. This often called a monad in BB. The reason is that a monad in Cat is the same as monad on a category.

Remarks on notation

It can be important to distinguish between a kk-tuply monoidal structure and the corresponding kk-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal GG-bundle in its groupoid incarnation. This is

G→EG→BG
G \to \mathbf{E}G \to \mathbf{B}G

(where EG=G//G\mathbf{E}G = G//G is the action groupoid of GG acting on itself). On the left we crucially have GG as a monoidal 0-category, on the right as a once-degenerate 1-category. Without this notation we cannot even write down the universal GG-bundle!

Or take the important difference between group representations and group 2-algebras, the former being functors BG→Vect\mathbf{B}G \to Vect, the latter functors G→VectG \to Vect. Both these are very important.

Or take an abelian group AA and a codomain like 2Vect2Vect. Then there are 3 different things we can sensibly consider, namely 2-functors

A→2Vect
A \to 2Vect

BA→2Vect
\mathbf{B}A \to 2Vect

and

B2A→2Vect.
\mathbf{B}^2A \to 2Vect
\,.

All of these concepts are different, and useful. The first one is an object in the group 3-algebra of AA. The second is a pseudo-representation of the group AA. The third is a representations of the 2-group BA\mathbf{B}A. We have notation to distinguish this, and we should use it.

Finally, writing BG\mathbf{B}G for the 1-object nn-groupoid version of an nn-monoid GG makes notation behave nicely with respect to nerves, because then realization bars |⋅||\cdot| simply commute with the BBs in the game: |BG|=B|G||\mathbf{B}G| = B|G|.

This behavior under nerves shows also that, generally, writing BG\mathbf{B}G gives the right intuition for what an expression means. For instance, what’s the “geometric” reason that a group representation is an arrow ρ:BG→Vect\rho : \mathbf{B}G \to Vect? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on BG\mathbf{B}G which is ρ\rho-associated to the universal GG-bundle:

the ρ\rho-associated vector bundle to the universal GG-bundle is, in its groupoid incarnations,

where VV is the vector space that ρ\rho is representing on, and this is classified by the representation ρ:BG→Vect\rho : \mathbf{B}G \to Vect in that this is the pullback of the universal VectVect-bundle

In summary, it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified should be, for instance concerning the crucial difference between the category in which GG lives and the 2-category in which BG\mathbf{B}G lives.